A157365 a(n) = 49*n^2 + 2*n.

51, 200, 447, 792, 1235, 1776, 2415, 3152, 3987, 4920, 5951, 7080, 8307, 9632, 11055, 12576, 14195, 15912, 17727, 19640, 21651, 23760, 25967, 28272, 30675, 33176, 35775, 38472, 41267, 44160, 47151, 50240, 53427, 56712, 60095, 63576, 67155

Offset

1,1

Comments

The identity (4802*n^2 + 196*n + 1)^2 - (49*n^2 + 2*n)*(686*n + 14)^2 = 1 can be written as A157367(n)^2 - a(n)*A157366(n)^2 = 1.

This formula is the case s=7 of the identity (2*s^4*n^2 + 4*s^2*n + 1)^2 - (s^2*n^2 + 2*n)*(2*s^3*n + 2*s)^2 = 1. - Bruno Berselli, Feb 11 2012

Also, the identity (49*n + 1)^2 - (49*n^2 + 2*n)*7^2 = 1 can be written as A158066(n)^2 - a(n)*7^2 = 1 (see Barbeau's paper in link). - Vincenzo Librandi, Feb 11 2012

The continued fraction expansion of sqrt(4*a(n)) is [14n; {3, 1, 1, 7n-1, 1, 1, 3, 28n}]. - Magus K. Chu, Sep 17 2022

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

E. J. Barbeau, Polynomial Excursions, Chapter 10: Diophantine equations (2010), pages 84-85 (row 15 in the first table at p. 85, case d(t) = t*(7^2*t+2)).

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

G.f.: x*(51+47*x)/(1-x)^3.

a(n) = 3*a(n-1) -3*a(n-2) +a(n-3).

E.g.f.: (51*x + 49*x^2)*exp(x). - G. C. Greubel, Feb 02 2018

Mathematica

LinearRecurrence[{3, -3, 1}, {51, 200, 447}, 50]

Prog

(Magma) I:=[51, 200, 447]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]];
(PARI) a(n)=49*n^2+2*n \\ Charles R Greathouse IV, Dec 23 2011

Crossrefs

Cf. A157366, A157367, A158066.

Sequence in context: A245829 A369922 A031431 * A157916 A007264 A158640

Adjacent sequences: A157362 A157363 A157364 * A157366 A157367 A157368

Keyword

nonn,easy

Author

Vincenzo Librandi, Feb 28 2009

Status

approved